Decoupling Inequalities for the Tail Probabilities of Multivariate U-statistics
| dc.creator | de la Peña, Victor H. | |
| dc.creator | Montgomery-Smith, Stephen J. | |
| dc.date | 1993-09-13 | |
| dc.date | 1999-12-06 | |
| dc.date.accessioned | 2026-07-07T09:04:41Z | |
| dc.date.available | 2026-07-07T09:04:41Z | |
| dc.description | In this paper the following result, which allows one to decouple U-Statistics in tail probability, is proved in full generality. Theorem 1. Let $X_i$ be a sequence of independent random variables taking values in a measure space $S$, and let $f_{i_1...i_k}$ be measurable functions from $S^k$ to a Banach space $B$. Let $(X_i^{(j)})$ be independent copies of $(X_i)$. The following inequality holds for all $t \ge 0$ and all $n\ge 2$, $$ P(||\sum_{1\le i_1 \ne ... \ne i_k \le n} f_{i_1 ... i_k}(X_{i_1},...,X_{i_k}) || \ge t) \qquad\qquad$$ $$ \qquad\qquad\le C_k P(C_k||\sum_{1\le i_1 \ne ... \ne i_k \le n} f_{i_1 ... i_k}(X_{i_1}^{(1)},...,X_{i_k}^{(k)}) || \ge t) .$$ Furthermore, the reverse inequality also holds in the case that the functions $\{f_{i_1... i_k}\}$ satisfy the symmetry condition $$ f_{i_1 ... i_k}(X_{i_1},...,X_{i_k}) = f_{i_{π(1)} ... i_{π(k)}}(X_{i_{π(1)}},...,X_{i_{π(k)}}) $$ for all permutations $π$ of $\{1,...,k\}$. Note that the expression $i_1 \ne ... \ne i_k$ means that $i_r \ne i_s$ for $r\ne s$. Also, $C_k$ is a constant that depends only on $k$. | |
| dc.identifier | https://arxiv.org/abs/math/9309211 | |
| dc.identifier | http://arxiv.org/abs/math/9309211 | |
| dc.identifier | Annals Prob. 23, (1995), 806-816 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/149469 | |
| dc.subject | Functional Analysis | |
| dc.subject | 60E | |
| dc.title | Decoupling Inequalities for the Tail Probabilities of Multivariate U-statistics | |
| dc.type | text |